Development of Pervious Concrete Pavement Performance Models Using Expert Opinions

Development of Pervious Concrete PavementPerformance Models Using Expert Opinions

Amir Golroo, Ph.D., P.Eng.1; and Susan L. Tighe, Ph.D., P. Eng.2

Abstract: Pervious concrete pavement (PCP) is of significant importance in the field of stormwater management in terms of reducing runoffvolume. Stormwater managers should initially ensure that PCP adequately performs over time to be able to implement it in a stormwatermanagement system. Performance models are intended to predict the performance of an asset over its service life. To develop a performancemodel commonly long-term performance data are essential. No performance model has been developed for PCP to date because PCP long-term performance data are rarely available. In such a case, expert knowledge is an alternative method to collect data for developing aperformance model. This research aims to develop performance models for PCP for the first time by using an integrated Markov chaintechnique (combination of homogenous and non-homogenous techniques) through incorporation of expert knowledge. Both deterministicand stochastic approaches are applied to build up Markov models by using expected values and the Latin hypercube simulation technique,respectively. Both approaches provide consistent results although the stochastic Markov model provides more detailed results. Short-termexperimental field data are also incorporated to validate the Markov performance models. DOI: 10.1061/(ASCE)TE.1943-5436.0000356.© 2012 American Society of Civil Engineers.

CE Database subject headings: Concrete pavements; Performance characteristics; Stormwater management; Models.

Author keywords: Performance model; Expert opinion; Pervious concrete pavement.

Introduction

Probabilistic performance models present the future condition ofpavement by using a mean, standard deviation, and appropriateprobability distribution functions. There are four common typesof probabilistic models commonly applied in developing pavementperformance models that use probabilistic tools (Lytton 1987):Markov models, Bayesian regression (Hajek and Bradbury 1996;Molzer et al. 2001), survivor curves, and semi-Markov models(Golroo and Tighe 2009). The main advantage of using a probabi-listic approach combined with the other tools is to consider uncer-tainty in pavement performance prediction. This approach betterdescribes the real world as compared with the deterministic ap-proach. The other advantage of probabilistic models, particularlythe Markov chain, is the ability to handle an incomplete, low qual-ity, and imprecise database by incorporating expert knowledge.These models include mechanisms to adjust prediction modelsonce complete performance data become available.

Pervious Concrete Pavement (PCP) is a newer product toaccommodate growing environmental demands. The PCP is oneof the best stormwater management practices according to the

Environmental Protection Agency (EPA) (Tennis et al. 2004).The PCP can overcome several obstacles with which stormwatermanagers are dealing (e.g., runoff). To better utilize PCP, its per-formance should be investigated over its service life. There has notbeen a clear understanding of PCP performance over time to date,resulting in limited usage of PCP, particularly in colder climates.

To investigate asset performance over time, long-term perfor-mance data are commonly applied, which has been rarely availablefor PCP. Thus, the expert knowledge would be an adequate alterna-tive to produce a performance model for PCP. Markov chain is amethod that applies expert knowledge to build a performance modelwithout having long-term performance data. The obtained data,however, should represent the reality. In fact, several research studieshave shown that this process has been adequately accomplished fordifferent types of pavement (Karan 1977; Li et al. 1996; Ortiz-Garciaet al. 2006). Golabi and Shepard (1997) developed a maintenance/improvement system for U.S. bridge networks. They appliedMarkovian models and mathematical programming tools to developdeterioration models. Kulkarni (1985) applied expert opinions ofpavement engineers together with field supervisors in developingperformance models for different pavement attributes (e.g., environ-mental conditions and traffic loads). The author utilized this ap-proach to establish a pavement management system for theKansas Department of Transportation. In the Markov chain process,the future condition of pavement is predicted solely by using the cur-rent condition of pavement (Hass 1997). A transition probabilityma-trix (TPM) describes the probability that an asset in a currentcondition shifts to a future condition. A drawback of the Markovchain process is the need for developing a TPM for each asset clas-sification. (A combination of factors affect asset performance.) Assetperformance history, also, cannot be captured in the Markov chainprocess because the future condition of an asset only depends on thecurrent condition. To tackle the latter problem, experimental realworld data have also been incorporated in this research to validateMarkov performance models. In short, the combination of Markov

1Assistant Professor, Amirkabir Univ. of Technology, Hafez Ave.,Tehran, Iran; formerly, Research Associate, Dept. of Civil and Environmen-tal Engineering, Univ. of Waterloo, Waterloo, ON, Canada N2L 3G1(corresponding author). E-mail: [email protected]

2Professor and Canada Research Chair, Norman W. McLeod Professorin Sustainable Pavement Engineering, Director of Centre for Pavement andTransportation Technology, Dept. of Civil and Environmental Engineering,Univ. of Waterloo, Waterloo, ON, Canada N2L 3G1. E-mail: [email protected]

Note. This manuscript was submitted on March 29, 2011; approved onSeptember 12, 2011; published online on September 14, 2011. Discussionperiod open until October 1, 2012; separate discussions must be submittedfor individual papers. This paper is part of the Journal of TransportationEngineering, Vol. 138, No. 5, May 1, 2012. ©ASCE, ISSN 0733-947X/2012/5-634–648/$25.00.

634 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / MAY 2012

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http://dx.doi.org/10.1061/(ASCE)TE.1943-5436.0000356




chain process and experimental data has been found as an efficientapproach for developing a performance model.

Objective

The main objective of this research is to develop performancemodels (deterministic and stochastic) for PCP by using the inte-grated Markov chain process (combination of homogenous andnon-homogenous) through incorporation of expert opinions.

The scope of this research is to produce an extensive tool that fol-lows an easy process to develop Markov models for PCP. The processaims to conduct a survey to collect expert opinions on PCP deteriora-tion probabilities for building TPMs. As a result, TPMs are used todevelop both deterministic and stochastic Markov models by usingthe expected value (EV) concept and Latin hypercube simulation(LHS) technique, respectively. Finally, short-term experimental fielddata are also incorporated to validate theMarkov performancemodels.

Research Approach

After a thorough literature review, it has been realized that there is stilla gap in performancemodeling of PCP because of a lack of long-termperformance data. Therefore, expert knowledge has been used as adata source to developMarkovmodels. A questionnairewas designedon the basis of different PCP categories (in terms of pervious concretethickness, traffic load, pavement age, and climate condition) and atraining session was primarily held. Then, 14 experts were selectedfor rating and given the questionnaire. Their mean panel ratings wereapplied to set up TPMs in deterministic approach, whereas, in thestochastic approach, a probability distribution function (PDF) was as-signed to mean panel ratings. After setting up TPMs, their elementswere validated through an appropriate validation approach, i.e., datasplitting. Namely, it set aside a few responses and applied them toinvestigate that there was no significant difference between these re-sponses and those remaining. For this purpose, additional question-naires were distributed to highly experienced engineers and theirresponses were applied to validate TPMs. After that, the PCP condi-tion was predicted by using TPMs and a current probability conditionvector of PCP through application of the Markov chain process. Toestimate the future condition of PCP, the concept of EVwas applied inthe deterministic approach, whereas, the LHS technique was utilizedin the stochastic approach. Short-term, real-world performance datawere applied to validate the Markov performance models. The futurework is to broaden this research by applying long-term performancedata to calibrate the Markov performance models.

Markov Chain Process Review

Themost important part of theMarkov chain process is to set up TPMsthat describe conditional probabilities. TMPs are the probabilities thatanasset inagivenconditionatagiventimewill shift toanotherconditionlevelorremaininthesameconditionlevel inastage.Conditionlevelsarealsodefinedasa“state”(e.g.,verygood,good)anda“stage” isdescribedas one year of traffic and environmental degradation. Three restrictionsshouldbeaccommodatedtodevelopMarkovmodels(Ortiz-Garciaetal.2006).TheMarkovmodel shouldhavea finite state space,bediscrete intime, and satisfy theMarkovian property. TheMarkovian property is tostate that the conditional probability of any future events, given anypastevents and the present state, is independent of the past events and de-pends only on a present state (Hillier and Lieberman 1990). Mishalaniand Madanat (2002) tested the Markovian properties of their stocha-stic duration models applied for infrastructure maintenance and

rehabilitation decision-making tools. The conditional probability forthe process to shift from state i in stage t to state j in stage t þ 1 is calledthe transition/conditional probability “pij” given by Eq. (1) as:

pij ¼ prob½Xðt þ 1Þ ¼ j∕XðtÞ ¼ i� ð1Þ

inwhichpij =theconditionalprobability starting fromstate i andendinginstate jafteronestage.Namely,pij showstheprobabilityof transitiontostate j (in stage t þ 1)given that thecurrent state is i (in stage t).TheXðtÞis a function of time, which expresses the condition of a system in aparticular point of time. The pij should meet the following constraints(Wang et al. 1994):

0 ≤ pij ≤ 1; for all i and j; i; j ¼ 0; 1; 2;…;M ð2ÞXMj¼0

pij ¼ 1; for all i; i ¼ 0; 1; 2;…;M ð3Þ

inwhich iand jaredefinedwithin theM-statespace.M =totalnumberofstates. These transition probabilities can be arranged in a one-stepTPMas follows:

TPMI ¼p11 p12 L p1Mp21 p22 L p2MM M M MpM1 pM2 L pMM

2664

3775

TPMI is the case with the deterioration and improvement (main-tenance). For instance, p21 is the probability that a pavement sectionshifts (improves) from state two to state one after one stage. Thistransition occurs only when a maintenance action is carried out,whereas TPMD could show the deterioration only (as follows).In this type of TPM, Pij is equal to zero for each i is greater thanj and pMM (probability of staying in stateM after one stage) is equalto one with regard to Eq. (3). Only the effects of deterioration on apavement section are considered in this paper.

TPMD ¼

p11 p12 p13 L p1M0 p22 p23 L p2M0 0 p33 L p3MM M M M M0 0 0 L 1

266664

377775

TheMarkov chain process incorporates both TPM and probabilitycondition vector [PðtÞ] to predict the future condition of an asset. Forexample, assume that the condition levels range from 1 to 5, and it isdivided into four states: 5-4, 4-3, 3-2, and 2-1. Suppose that the prob-ability condition vector of an asset at stage “t” would lie within theinterval of 4-3 and 3-2with probability of 80% and 20%, respectively.Therefore,PðtÞ canbepresented as (0, 0.8, 0.2, 0). The condition of anasset in the next stage can bepredicted byusing the current probabilitycondition vector and TPMs, as follows:

PðtÞ ¼ Pðt � 1Þ × TPMt ð4Þin which PðtÞ = probability condition vector of an asset at stage t,Pðt � 1Þ = probability condition vector of an asset at stage t � 1,and TPMt = transition probability matrix corresponding to transitionfrom stage t � 1 to t.

Thus, the probability condition vector of an asset at any particulartime can be specified byusing an initial probability conditionvector ofthe asset and TPMs associated with stages 1 to t, as follows:

PðtÞ ¼ Pð0Þ × TPM1 × TPM2 × L × TPMt ¼ Pð0Þ ×Yti¼1

TPMi

ð5Þ

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inwhichPð0Þ= initial probability conditionvector of an asset (at staget ¼ 0), TPMi = transition probability matrix corresponding to transi-tion from stage i� 1 to i (i ¼ 1; 2;…; t), andΠ (Pi) = the product of asequence of terms that derives from the capital letter in the Greekalphabet.

Two main approaches have been used by researchers to developMarkov models to date: homogenous and non-homogeneous.The homogeneous Markov chain applies identical TPM for allstages. In other words, the series of transition probabilities of trans-ferring from one state to another is time-independent. Chapman-Kolmogorov relations provide a simplified version of Eq. (4)presented as follows (Hillier and Lieberman 1990):

PðtÞ ¼ Pð0Þ × TPM1 × TPM1 × L × TPMt

¼ Pð0Þ × TPM × TPM × L × TPM ¼ Pð0Þ × TPMt ð6Þ

in which Pð0Þ = initial probability condition vector of an asset(at stage t ¼ 0), TPMi= transition probability matrix correspondingto transition from stage i� 1 to i (i ¼ 1; 2;…; t), and TPM =individual transition probability matrix corresponding to transitionfrom a stage to the next stage.

According to this approach, the probability condition vector atstage “t” can be obtained by product of the initial probability con-dition vector Pð0Þ and TPM to the power of “t”. However, anon-homogeneous Markov chain process is more realistic but morecomputationally complex. In this case, TPMs are time-dependentand would change at any stages during the service life of an assetbecause of changing in traffic volumes, subgrade strength, aging,and climate conditions. An ideal approach is to develop non-homogenous TPMs. However, this approach requires much morerelated knowledge and experience than the homogenous approach.Beside, it is hardly feasible to build non-homogeneous TPMs for allstages over a planning horizon in the case of assets that suffer fromlong-term performance data limitation such as PCP. The combina-tion of non-homogeneous and homogeneous approaches is an effi-cient technique to take into account changes that may occur in termsof deterioration rate and at the same time does not require long-termperformance data.

Markov Chain Survey

Two main methods have been used in the related literature to de-velop TPMs. These methods are applying subjective data and uti-lizing long-term performance data (Karan 1977; Li et al. 1996;Ortiz-Garcia et al. 2006). The latter approach is not applicable tothis study because of limited knowledge of long-term performanceof PCP. Consequently, the subjective approach was selected tobuild TPMs for various groups employing a panel of experiencedengineers including 14 experts. Note that a limited number of ex-perts have participated in this survey because of the lack of exper-tise in the field of PCP performance evaluation. However, theperformance model can be readily updated once long-term perfor-mance data is available because of the flexibility and compatibilityof the Markov chain process.

After extensive discussions with experts, a few questionnaireswere designed and pilot tests were carried out. On the basis of

the feedback of participants, the questionnaires were adjusted anda final version of the questionnaire was developed. Appendix Ishows the questionnaire that was distributed to experts in this study.The questionnaire included a brief explanation on the concept ofthe PCP condition index and the Markov chain process, and a seriesof TPMs for various PCP groups were presented. These groupswere categorized on the basis of PCP characteristics (pavementage, pervious concrete thickness, traffic load, and climate condi-tion). The PCP characteristics are summarized in Table 1.

The most common pervious concrete thickness being used bydesigners is 150 mm (6 in.) according to site visits and related lit-erature. So, 150 mm was used as a trigger level between thin andthick PCP.

The “Light” traffic was assigned to a pavement section that isgenerally exposed to ordinary cars, vans, and trucks. The heavyvehicle is limited to trucks that have at least six wheels excludingpanel and pickup trucks. The “Heavy” traffic is described as a pave-ment section that is essentially exposed to heavy vehicles (such as aplant, which is frequently exposed to heavy vehicles with morethan 25 average daily truck traffic).

Two intervals were applied: primary and secondary. The pri-mary interval covers a shorter time period because the PCP deterio-ration rate is higher during the first few years rather than later years.Therefore, a shorter primary interval was used to keep track ofdeterioration in more detail. Also, an analysis period of five years(a short time period) was selected because of a lack of long-termrelated experience. The interval classifications are consistent withfield investigations and real-world performance of PCP. Raveling isthe major issue because of two attributes of PCP: the lack of fineaggregates and a low water-to-cement ratio. These two attributesare too limited in PCP to have sufficient porosity (permeability).It has been observed that the most dominant distress in the earlierage of PCP is raveling. The substantial amount of raveling happensright after installation once traffic starts loading on PCP.

The “Hard wet freeze” climate condition was defined as certainwet freeze areas that undergo a number of freeze-thaw cycles an-nually (15þ) and there is precipitation during the winter in whichthe ground stays frozen as a result of a long continuous period ofaverage daily temperatures below freezing. These areas would havesituations in which PCP becomes fully saturated.

Two levels of pervious concrete thickness, two traffic loadpatterns, two levels of pavement age, and one type of climatecondition were used, resulting in eight (2 × 2 × 2 × 1) possiblecombinations, and eight possible pavement groups. Because a typ-ical design has been used for PCP, groups that have thin perviousconcrete thickness and heavy traffic are rarely feasible. Therefore,these groups were eliminated and at a total of six groups wereanalyzed (Table 2).

Before distributing the questionnaire, respondents were trained.The Markov chain process was explained and the method of com-pleting TPMs was discussed. Several photos were provided show-ing the condition state of PCP [pervious concrete distress index(PCDI) defined subsequently] of several sections during their ser-vice life. Having trained the respondents, the questionnaire was dis-tributed to them and they completed and returned the survey.A sample completed TPM is shown in Table 3.

Table 1. PCP Group Characteristics

Pervious concrete thickness Vehicle traffic Pavement age Climate condition

Thin 100mm < H ≤ 150mm (4 in: < H ≤ 6 in:) Light Primary interval (1year < T ≤ 2year) Hard wet freeze

Thick 150mm < H < 250mm (6 in: < H < 10 in:) Heavy Secondary interval (2year < T < 5year)


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Each cell of the final TPMs is the adjusted average response ofthe panel of respondents. A theoretical constraint applied to TPMswas that the sum of cells located in each row should be equal to 1.0.However, the sum of average values of each row would not be nec-essarily equal to 1.0 [Eq. (3)]. Therefore, the sum of average valuesof each row was applied to adjust each average value so that sumof each row became 1.0. The adjustments were very small. Thefinal six TPMs and their standard deviations are presented inTables 21–32 in Appendix II. The TPMs have shown that thedeterioration rate of PCP in the first few years of age is higher thanthe deterioration rate of remaining service life, which makes sense.In essence, the deterioration rate slows down after an initial periodof time. Also, the TPMs have shown that the deterioration rates ofgroups 3 and 4 (thick pervious concrete thickness and light trafficload) were the lowest, whereas that of groups 5 and 6 (thick per-vious concrete thickness and heavy traffic load) were the highest,which is consistent with field investigations. This shows that trafficloading had the most significant effect on PCP condition.

The entire TPMs were tested to ensure they could successfullypredict expert opinions; however, actual long-term field perfor-mance data is required to sufficiently validate the models. For thispurpose, the data splitting technique was used, i.e., 75% of data wasapplied as a database for setting up TPMs and 25% of data was usedfor testing. A statistical test was conducted to examine whether ornot a significant difference existed between these two sets of data.T-tests (two cases: σ1 ¼ σ2 and unknown, σ1 ≠ σ2 and unknown,σ ¼ sample mean) and F-test were applied for this purpose(Table 4). The null hypothesis (H0) was that the mean of the data-base was equal to that of testing data at the 95% confidence level,which was accepted. It was concluded that there was no significant

difference between two distributions. Consequently, the final TPMswere successfully tested.

Markov Chain Model Development

To develop a Markov model a condition state (condition index), aninitial condition probability vector, stages, and TPMs should be de-fined. A condition state and an initial condition probability vectorare defined in the next sections. (Stages and TPMs have been de-fined previously).

Condition State

As noted previously, PCDI was applied in this research as a con-dition index to develop TPMs. PCDI describes overall surface con-dition of PCP, which was defined in the questionnaire and trainingsession (along with some pictures). Surface distresses were evalu-ated, incorporating the adjusted Ministry of Transportation ofOntario (MTO) protocol. Adjustments involved in the pavementevaluation and weighting factors are further discussed. Distresstypes detected on PCP surfaces were assessed describing distressseverity and density. The pavement evaluation adjustment was thata percentage of density was applied to the various severity levels asopposed to assignment of 100% in only one level, which is com-monly carried out in practice. Namely, in a case in which two levelsof distress severity are present on a section, only one of them isrecorded in the MTO protocol. Thus, in this paper more detail datawas recorded. The PCDI was calculated by applying Eq. (6).

The weighting factor adjustments were to establish a new set ofdistress weighting factors by a group of experienced engineers forPCP. The weighting factors represent the relative effect of each dis-tress on PCDI. In fact, the PCP weighting factors are different fromweighting factors provided for conventional concrete by the MTOprotocol. Because the most dominant distress of PCP is raveling,the highest weight was dedicated to this distress of 3.0. For in-stance, cracking, stepping, and polishing, which have not been ob-served very frequently on PCP, obtained the lowest weightingfactors of 0.5.

PCDI ¼ 10 ×PCDImax �

P6i¼1 wi

P5j¼1ðSij × DijÞ

PCDImaxð7Þ

Table 2. Different Pavement Groups

Group Pervious concrete thickness Vehicle traffic Pavement age Climate condition

1 Thin Light Primary interval Hard wet freeze

2 Thin Light Secondary interval Hard wet freeze

3 Thick Light Primary interval Hard wet freeze

4 Thick Light Secondary interval Hard wet freeze

5 Thick Heavy Primary interval Hard wet freeze

6 Thick Heavy Secondary interval Hard wet freeze

Table 3. Sample Transition Probability Matrix

PCDI

Future condition

State 1 (Very good) State 2 (Good) State 3 (Fair) State 4 (Poor) State 5 (Very poor)

8–10 6–8 4–6 2–4 0–2

Present condition State 1 (Very good) 8–10 50 50 — — —State 2 (Good) 6–8 — 60 40 — —State 3 (Fair) 4–6 — — 65 35 —State 4 (Poor) 2–4 — — — 70 30

Table 4. T-test and F-test for Validation of Markov Chain Model

t-test(σ1 ¼ σ2 and unknown)

t-test(σ1 ≠ σ2 and unknown) F-test

S2p 218 ν 90 Fobserved 1.50

tα∕2 2.63 tα∕2 2.63 Fα∕2 1.62

dupper 7.91 dupper 7.92 H0 Accepted

dlower �7:95 dlower �7:96 NA NA

H0 Accepted H0 Accepted NA NA


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in which PCDI = pervious concrete distress index, PCDImax =the maximum theoretical value dedicated to an individual pavementdistress (i.e., 30), i = distress number, j = severity levels,wi = weighting factor of distress i ranging from 0.5 to 3.0 (shownin Table 5), Sij = severity level j of distress i measured on a scale of0.5 to 4, and Dij = density level j of distress imeasured on a scale of0 to 100 (percentage).

The PCDI ranges from 0 to 10, representing very poor to verygood surface overall condition, respectively. To define conditionstates, PCDI was divided into five intervals, as shown in Table 6.The average of each interval (EPCDI) is also presented in this table.

Although more condition states would provide more detailedTPMs, they simultaneously decrease precision (or increase vari-ance) of TPM elements. Namely, the probabilistic process suffersfrom a decrease of precision of data with an increase of conditionstates. That is, the higher the number of states, the lower amountprecise data would be collected from experts to build TPMs. Hence,this research suggests only five condition states to deal with thisuncertainty.

Moreover, it is assumed that a PCP section can shift only fromone state to a consecutive lower state. The PCP has been commonlyutilized in the light duty applications and not been imposed to ex-cessive traffic loads. Therefore, degrading more than one state(e.g., premature failure) rarely happens so that the assumptionmakes logical sense. This assumption also increases the level ofaccuracy because in each row of TPMs only two elements willbe present and respondents can determine the values with moreconfidence. Moreover, because comprehensive PCP field perfor-mance is not available, respondents have limited related experienceso that they cannot confidently predict a transition from one state tomultiple lower states. This assumption assists respondents in apply-ing their knowledge and experience appropriately to predict thoseconditions that most probably happen in the real world. Consider-ing multi-level state changes would increase the vagueness and un-clearness because respondents have not had sufficient relatedexperience.

The respondents were asked to indicate the probability of stay-ing in a single condition state, e.g., “Very Good” condition (p11)presented by Eq. (8). Likewise, the respondents could determinethe probability of transiting from one condition state to the lowercondition state, e.g., from condition state “Very Good” to “Good”

(p12) presented by Eq. (9). Because the summation of elements ineach row should be equal to 1, having one element in an individualrow, the other element can be readily calculated by using Eq. (10).Therefore, there is only one variable in an individual row that sig-nificantly increases the accuracy of TPMs.

pii ¼ prob½Xðt þ 1Þ ¼ i∕XðtÞ ¼ i� ð8Þ

pii�1 ¼ prob½Xðt þ 1Þ ¼ i� 1∕XðtÞ ¼ i� ð9Þ

pii ¼ 1� pii�1 ð10Þin which pii = probability of staying in state i over stage t, andpii�1 = probability of shifting from state i to state i� 1 overstage t

Consequently, TPM can be represented as follows:

TPM ¼

Very goodGoodFairPoor

Very good Good Fair Poor Very poorp11 p12 0 0 00 p22 p23 0 00 0 p33 p34 00 0 0 p44 p45

0BBBB@

1CCCCA

in which p11 = probability that a PCP section in state “Very good”would stay in the same state over one stage (i.e., one year) and p12 =probability that a PCP section in state “Very good” would deterio-rate to the lower state (i.e., “Good”) over one stage. Likewise, theother cells of TPM can be defined.

Initial Condition Probability Vector

The Markov chain model starts with a PCDI probability vectoraddressing the initial condition of a given PCP section. An initialPCDI probability vector of a new PCP section was assumed asPCDIð0Þ ¼ ð0:9; 0:1; 0; 0; 0Þ on the basis of the observation, re-lated experience, and relevant literature. In this case, there are 90%and 10% chances that a new PCP section immediately after instal-lation will be in “Very good” condition and “Good” condition,respectively.

Homogenous versus Non-homogenous TPM

The Markov chain process has been widely assumed to be homog-enous over time (Karan 1977; Ortiz-Garcia et al. 2006). That is,a single TPM is applied for all stages. However, as noted, itis more realistic to assign different TPMs to various years (non-homogeneous Markov chain), but this approach increases theimprecision and inaccuracy in results. A combination of thesemethods has been used in this paper, namely, two intervals havebeen assumed: primary (first and second years) and secondary in-tervals (third, fourth, and fifth years). In each interval, TPMs arehomogeneous, whereas TPMs associated with the primary intervalare different from those of the secondary interval. Field investiga-tions and real-world data provide evidence for non-homogeneity indeterioration. As noted previously, on the basis of the real PCP per-formance data, the rate of deterioration is higher in the first fewyears after installation as opposed to the latter years. Hence, a non-homogeneous Markov chain has been defined by shifting from theprimary to secondary intervals. This shift was restricted to a case inwhich the deterioration pattern is likely to change significantly.Because short-term performance data of PCP is available, ashort-term prediction period has been selected for developingTPMs. That is, respondents cannot sufficiently predict the perfor-mance of PCP on a long-term period (more than five years)according to available data.

Table 5. Weighting Factors

Distress # Distress type Weight

1 Cracking 0.5

2 Polishing 0.5

3 Stepping 0.5

4 Potholing 1.0

5 Joint and crack spalling 2.0

6 Raveling and coarse aggregate loss 3.0

Table 6. Definition of Condition States

i Condition state

PCDI boundaries

EPCDILower limit Upper limit

1 Very good 8 10 9

2 Good 6 8 7

3 Fair 4 6 5

4 Poor 2 4 3

5 Very poor 0 2 1


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Deterministic Markov Model

The concept of EV has been used in this research to develop deter-ministic Markov models for PCP. To estimate the future probabilityvector of PCDI, the initial PCDI probability vector and TPMs wereused by applying Eq. (11) as follows:

PCDIðtÞ ¼ PCDIð0Þ ×YTt¼1

TPMt t ¼ 1; 2;…; 5 ð11Þ

in which PCDIðtÞ = probability vector of PCDI at the end of stage t,PCDI(0) = initial probability vector of PCDI, TPMt = transitionprobability matrix associated with stage t, and T = planning horizonwhich is equal to five years.

The PCDI probability vector of groups 1 and 2 over the planninghorizon (five years) were calculated using Eq. (11) and are shownin the form of histogram in Fig. 1 for illustration. Namely, PCDI(0),PCDI(1), PCDI(2), PCDI(3), PCDI(4), and PCDI(5) represent thePCDI probability vector of groups 1 and 2 after installation, year 1,year 2, year 3, year 4, and year 5, respectively.

It is desirable to have a single value for PCDI rather than a prob-ability vector for further calculations. For this purpose, the expectedvalue of the PCDI probability vector can be computed usingEq. (12), as follows:

EVPCDIðtÞ ¼ EPCDI × PCDIðtÞ ð12Þ

in which EVPCDI = expected value of the PCDI probability vectorafter stage t, EPCDI = vector of average of various state boundaries(i.e., 9, 7, 5, 3, 1) (see Table 6), and PCDIðtÞ = PCDI probabilityvector at the end of stage t.

For instance, the condition vector of a new PCP section is (0.9,0.1, 0, 0, 0) so that the expected value of its condition vector isequal to 8.8 ½EVPCDIðtÞ ¼ 9 × 0:9þ 7 × 0:1þ 5 × 0:0þ 3 × 0:0þ1 × 0:0 ¼ 8:8�.

Having applied Eq. (12), the expected value of PCDI was com-puted for each group within a five-year period. Performance models(according to the Markov chain process) for different PCP groupsare plotted in Fig. 2. The variance of expected values of PCDI in-creases with an increase in pavement age. This fact could not beshown in Fig. 2 because the performance model presented in thisfigure is deterministic and shows only the mean values. However,the distribution of the PCDI mean values can depict the increasingvariance (Fig. 1). Fig. 2 shows that groups 3 and 4 (thick perviousconcrete thickness and light traffic) reflect better predicted perfor-mance than the others, whereas groups 5 and 6 (thick pervious con-crete thickness and heavy traffic) have the worst performance. It isdeemed that heavy traffic has a more significant effect on PCDIrather than pervious concrete thickness, which matches with relatedfield experience. This shows that the survey results in transitionprobabilities, which make engineering sense.

Performance models characteristics have been compared inTable 7. Table 7 shows that all models are statistically significantwith high R2 values and a low standard error of estimate.

Stochastic Markov Model

The Markov chain method has been applied in this study in bothdeterministic and stochastic ways, whereas some researchers ap-plied the Markov chain method only in a deterministic way (Karan1977; Tighe 1997). As noted, the deterministic approach providesthe mean of panel ratings as elements of TPMs. However, a more

Fig. 1. Histogram of the PCDI probability vector over the planning horizon


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detailed approach is to fit a PDF to each element of TPMs to ex-press a real distribution of panel ratings. For this purpose, severalattempts have been made to fit an appropriate PDF (normal, expo-nential, gamma, and lognormal) to TPMs’ elements of variousgroups. The PDF, which had the best goodness of fit to the data(TPMs’ elements) has been selected through various methods(chi-square, Anderson-Darling, and Kolmogorov-Smirnov) by us-ing @Risk software (Palisade Corporation 2005). For instance, theGauss distribution function (Fig. 3) had the best goodness of fit tothe response values of p23 (probability of transition from state“Good” to state “Fair”) of group 5.

The stochastic variables, i.e., TPMt [(which includes probabilitydistribution functions) should be used in the calculation of PCDI(Eq. (11)]. For this purpose, the Latin Hypercube Simulation LHStechnique was applied. The simulation was performed 1,000 timesusing the @Risk software. The outcome of Eq. (11) (i.e., PCDI)was a histogram of results (not a single point estimate value). Sev-eral attempts have been made to fit an adequate PDF to the histo-gram of results [i.e., PCDIðtÞ] for all PCP groups (Table 8). Forinstance, the best fitted PDF to the histogram of results of PCDI(3) for Group 2 is shown in Fig. 4 for illustration.

It is desirable to plot a stochastic performance model to illustratethe deterioration trends of PCP groups over time. For this purpose,the mean value of PDF of each PCDIðtÞ was calculated togetherwith the mean values plus and minus standard deviation (approx-imately 70% of true mean is restricted to this domain). These valuesare presented for all groups in Table 8.

The stochastic performance curves for various PCP groups areillustrated in Fig. 5 by using data presented in Table 8. This figureexhibits that the variance of results in the first interval (first andsecond years) is less than the remaining years, which has a practicalsense. Moreover, the wider the prediction period, the more impre-cise the results are. That is, the range presented for PCDI(5) iswider (higher variance) than PCDI(4). This may not be distin-guished in Fig. 5; however, values presented in Table 8 can clearlyshow that variance increases with increase in pavement age(prediction period).

As it was anticipated, the overall performance of groups 3 and 4(light traffic and thick pervious concrete thickness) was the best,whereas that of groups 5 and 6 (heavy traffic load and thickpervious concrete thickness) was the worst. In addition, the mostconsistent results (higher precision, i.e., lower variance) were re-lated to groups 1 and 2, which is reasonable because groups 1and 2 have been widely used (the most common PCP design)and the respondents are more familiar with their performance asopposed to the other PCP groups, which have been moderately uti-lized in the real world.

Moreover, stochastic Markov models for various groups havebeen developed on the basis of mean values represented in Table 8using regression analysis. Table 9 summarizes the characteristics ofthese models. This table shows that all models were fitted to the

Fig. 2. PCP performance models using Markov chain

InvGauss(139.76; 24505.13) Shift=-99.688

Val

ues

x 10

^-2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

15 20 25 30 35 40 45 50 55 60 65

< >48.8%51.1%10.60 40.00

Fig. 3. The best PDF fitted to panel’s responses associated with p23 ofgroup 5

Table 7. Deterministic Markov Models for PCP

Model Independent variable B Std. error t p-value R2 Std. error of the estimate

Groups 1 and 2 (Constant) 8.75 0.030 296 0.000 0.999 0.0172

Age �0:72 0.010 �74 0.000


Age �0:56 0.006 �99 0.000


Age �0:83 0.003 �720 0.000


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data adequately. The models have excellent R2 values and a lowstandard error of the estimate.

Ultimately, the results achieved through incorporation of bothMarkov chain approaches (deterministic and stochastic) were ap-proximately the same (compare Figs. 2 and 5 or Tables 7 and 9).However, the stochastic Markov model provides more detailed re-sults than the deterministic one.

Performance models are usually applied to predict service life ofassets. To predict the service life of PCP, stochastic models havebeen utilized. To determine the service life of PCP, the minimumacceptable level of PCDI should be indicated. The acceptable levelwas assumed to be 3.0 in this research because PCP is commonlyused on low volume roads and parking lots. The PCP overall

condition in parking lots and low volume roads does not signifi-cantly affect traffic flow, speed, and safety. Therefore, a lower valuewas selected for the PCDI acceptable level as opposed to other ap-plications such as high volume roads (e.g., highways). The servicelife of PCP using stochastic models presented in Fig. 5 was esti-mated to be approximately 10, 14, and 8 years for groups 1 and 2, 3and 4, and 5 and 6, respectively. Namely, these graphs (mean val-ues) were extrapolated to reach the minimum acceptable level(PCDI ¼ 3), then the associated points in time were consideredas approximate service life of corresponding groups. These servicelives are only estimated values (regression extrapolation cannotprovide an accurate prediction) and should be validated in the fu-ture by using in-service data. However, the estimate of 8 to 14 yearsfor PCP service life does seem reasonable given current perfor-mance levels.

LogLogistic(-0.91302; 7.5816; 7.9011)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

2 4 6 8 10 12 14 16

< >5.0% 5.0%90.0%4.31 10.09

Fig. 4. The best PDF fitted to the histogram of results of PCDI(3) ofgroup 2

Table 8. Best Fitted PDFs to PCDIðtÞ for All GroupsGroup Age (Year) PCDIðtÞ PDF fitted to PCDIðtÞ Mean std. Meanþ std: Mean� std:

1 1 PCDI(1) Loglogistic (�8:4777, 16.545, 26.772) 8.1 1.1 9.2 7

1 2 PCDI (2) Loglogistic (�1:5738, 8.9144, 9.222) 7.5 1.8 9.3 5.7

2 3 PCDI (3) Loglogistic (�0:91302, 7.5816, 7.9011) 6.9 1.8 8.7 5

2 4 PCDI (4) Pearson 5 (26.644, 242.34, risk shift (�3:1739)) 6.3 1.9 8.2 4.4


3 1 PCDI(1) Loglogistic (0.86295, 7.2802, 9.8461) 8.3 1.4 9.7 6.9

3 2 PCDI (2) Loglogistic (�1:0026, 8.5557, 6.9203) 7.9 2.4 10 5.4



4 5 PCDI (5) Loglogistic (0.31894, 5.7881, 3.9414) 6.8 3.4 10 3.3

5 1 PCDI(1) Logistic(7.9645, 0.93792) 8.0 1.7 9.7 6.3


6 3 PCDI (3) Loglogistic (�7:4777, 13.85, 10.568) 6.6 2.5 9 4.1



Note: “Std.” = standard deviation.

Fig. 5. Stochastic performance curves of various PCP groups


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Deterministic versus Stochastic Markov Models

It is desirable to compare the performance models that were devel-oped using deterministic and stochastic approaches in more details.For this purpose, outcomes of each model were compared for theentire groups over the planning horizon. The t-test was conductedto check whether or not there was a significant difference betweenoutcomes of stochastic and deterministic models. The results of t-test showed that two models’ outcomes did not have statisticallysignificant difference at the 95% level of confidence (Table 10).

Stochastic and deterministic models were plotted separately forthe entire groups in pairs (1 and 2, 3 and 4, and 5 and 6). Fig. 6illustrates that both models (i.e., deterministic and stochastic) pro-vide consistent results over the prediction period (five years). How-ever, in all cases, the stochastic models provide more conservativeresults than the deterministic models. Moreover, this figure showsthat groups 5 and 6 deteriorate faster than the others, whereasgroups 3 and 4 degrade slower among the rest of the groups, whichis consistent with field observations and real-world performanceof PCP.

Model Validation

To validate the experts’ opinions and models developed on the basisof their ratings, some field experiments have been conducted.

For this purpose, 11 PCP parking lots, which are located in the stateof Ohio in the United States, were investigated. These parking lotswere carefully selected and evaluated considering the fact that theyare all located in the hard wet freeze condition. The entire PCPsections were evaluated on the basis of PCDI and plotted againstpavement age (Fig. 7). As shown on Fig. 7, a linear model is fittedto PCDI data with a good value of R2. This figure presents anexperimental performance model that should be compared withthe developed Markov models.

Most of the PCP sections have thin concrete thickness and lighttraffic load so that they could be compared with groups 1 and 2.They have been compared by using t-test to examine whether or notthere is any significant difference between experimental data andexperts’ opinions (deterministic and stochastic models). The resultof t-test shows that the Markov models’ outcomes are not statisti-cally different at the 95% of confidence level from the experimentalmodel (Table 11).

Conclusions

The Markov chain process was applied in this research to developperformance models for PCP by using expert knowledge becauselong-term performance data has not been thoroughly available.Both deterministic and stochastic approaches were utilized insetting up Markov models, which provided consistent results.The Markov models have been successfully validated by using

Table 9. Stochastic Markov Models for PCP

Model Independent variable B Std. error t p-value R2 Std. error of the estimate


Age �0:60 0.014 �43 0.000


Age �0:41 0.023 �17 0.000


Age �0:72 0.011 �66 0.000

Table 10. T-test and F-test for Deterministic and Stochastic MarkovModels

t-test(σ1 ¼ σ2 and unknown)

t-test(σ1σ2 and unknown) F-test

S2p 1.42 ν 33.00 Fobserved 0.70

tα∕2 2.72 tα∕2 2.73 Fα∕2 2.27

dupper 1.37 dupper 1.37 H0 Accepted

dlower �0:79 dlower �0:80 NA NA

H0 Accepted H0 Accepted NA NA

Fig. 6. Stochastic versus deterministic models for various PCP groups

Fig. 7. Experimental performance model


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short-term experimental data. The following conclusions have beenderived:1. The Markov chain models showed that the variance of the pre-

dicted PCP condition index increased with increase in pave-ment age over the planning horizon.

2. The TPMs showed that the predicted deterioration rate of PCPin the first few years of age was higher than that of remainingservice life. In essence, the predicted deterioration rate sloweddown after an initial period of time.

3. The TPMs showed that the predicted deterioration rates ofgroups 3 and 4 (thick pervious concrete thickness and lighttraffic load) were the lowest, whereas that of groups 5 and6 (thick pervious concrete thickness and heavy traffic load)were the highest. The predicted deterioration rates of Groups1 and 2 (thin pervious concrete thickness and light traffic load)were between the others. This shows that traffic loading hadthe most significant effect on PCP condition.

4. The most precise results (less variance) were related to groups1 and 2, which were reasonable because groups 1 and 2 havebeen widely used (the most common PCP design) and therespondents were more familiar with their performance as op-posed to the other PCP groups, which have been moderatelyutilized in the real world.

5. Deterministic and stochastic models provided approximatelythe same results over the prediction period (five years). How-ever, in all cases, the stochastic models provided more conser-vative results than the deterministic models.

6. The service lives of PCP using stochastic models were pre-dicted to be approximately 10, 14, and 8 years for groups 1and 2, 3 and 4, and 5 and 6, respectively. It should be notedthat these numbers are only estimated values (regression extra-polation cannot provide an accurate prediction) and should bevalidated in the future by using in-service data. However, theestimate of 8 to 14 years does seem reasonable given currentperformance levels.

7. The short-term experimental data that was collected from11 PCP parking lots provided consistent results with theMarkov models. The difference between outcomes of the ex-perimental model and the Markov models were not statisticallysignificant.

Appendix I. Developing Markov Models

Pavement States

The performance model predicts the condition index of a pavementat any specific time. PCDI as defined in this paper addresses anoverall condition of a pavement affected by surface distresses (in-cluding raveling, spalling, cracking, potholing, polishing, and step-ping. The total range of PCDI (0–10) is divided into five discreteranges (i.e., Very good, Good, Fair, poor, and Very Poor), each

expressing a state, as shown in Table 12, 0 shows a pavement thatunmistakably requires a repair action whereas 10 presents a pave-ment like new.

Pavement Groups

A pervious concrete pavement group is defined by the combinationof specific attributes: pervious concrete thickness, climate condi-tion, pavement age, and traffic load. Two levels of pavement thick-ness, one type of climate condition (i.e., hard wet freeze climatesuch as North Ontario, Canada), two levels of pavement age,and two traffic load patterns are used, resulting in eight ð2 × 2 ×2 × 1Þ possible combinations, and 8 possible pavement groups.Regarding the fact that a typical design has been used for perviousconcrete pavements, groups that have thin overlay and heavy trafficare rarely available. Therefore, the associated groups have beeneliminated and six groups are analyzed. Table 13 shows the perfor-mance factors and their levels, and the six most feasible combina-tions/groups. The implicit assumption is that a change in the levelof each previously noted factor leads to a significant change in thepavement condition index.

Transition Probability Matrices

This section expresses how a specific group of pavement currentlyin a particular state will change (i.e., make a “transition”) to thelower state or remain in the same state after one year of serviceassuming that no maintenance action has been carried out. You willbe asked to fill out one table for each pavement group on the basisof your own experience. The numbers you report should expressyour opinion that a pavement that now occupies a specific statewill stay in the same state or will degrade to the immediate lowerstate at the end of one year. It is assumed that a pavement cannotdegrade more than one state. For instance, a pavement in state 4cannot degrade to state 2 or 1 after one year of service. A similarconclusion can be extended to other states.

There will be one matrix with five columns and four rows foreach pavement group. The states on the left hand side of the table(row) specify the present state of the pavement and the states alongthe top of the matrix (column) present possible states after one yearof service. Table 14 shows an example transition matrix.

If we represent any box in Table 14 by Pij then this Pij representsthe number of pavement sections out of 100 of the same group withinitial state i that would be expected to be in state j at the end of oneyear assuming under a “do nothing” treatment alternative. For in-stance, if you think that 35 out of one hundred pavement sections ingroup 3, whose initial state is 3 (PCDI: 4–6), would degrade to the

Table 11. T-test for Experimental, Deterministic, and Stochastic Markov Models

Experimental versus deterministic model Experimental versus stochastic model

t-test (σ1 ¼ σ2 and unknown) t-test (σ1σ2 and unknown) t-test (σ1 ¼ σ2 and unknown) t-test (σ1σ2 and unknown)

S2p 1.18 ν 8.00 S2p 0.88 ν 8.00

tα∕2 3.11 tα∕2 3.36 tα∕2 3.11 tα∕2 3.36

dupper 1.02 dupper 1.18 dupper 1.04 dupper 1.17

dlower �2:87 dlower �3:02 dlower �2:33 dlower �2:47

H0 Accept H0 Accept H0 Accept H0 Accept

Table 12. PCDI of Different States

State 1 (Very good) 2 (Good) 3 (Fair) 4 (Poor) 5 (Very poor)

PCDI 8–10 6–8 4–6 2–4 0–2


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lower state 4 (PCDI: 2–4) at the end of one year, then P34 ¼ 35 andP33 ¼ 65 as shown in Table 14.

We asked you to apply the following procedure for filling in theblank tables provided for each pavement group.1. Please read the title of each table to familiarize yourself with

the pavement group described.2. Start with the top row (PCDI: 8–10) and ask yourself the fol-

lowing questions:If I had 100 pavements of this class in state 1 (PCDI: 8–10)

howmanyof themwould Iexpect to stay instate1 (PCDI:8–10)?How many of them would degrade to state 2 (PCDI: 6–8) afterone year. Your answers should go in the appropriate cells.

Note You do not have to fill all the cells in a row. You areonly asked to fill two cells in each row which are the proba-bility of staying in the same state (diagonal cells) and the prob-ability of degrading into the immediate lower state (cells nextto diagonal cells).

3. When you finish the top row, please go to the second row(PCDI: 6–8) and ask yourself the following questions:

If I had 100 pavements of this class in state 2 (PCDI: 6–8)how many of them would I expect to stay in state 2 (PCDI:

6–8)? How many of them would degrade to state 3 (PCDI:4–6) after one year. Your answers should go in the appropriatecells.

Note that a pavement cannot be improved (i.e., change to ahigher state) because no maintenance action has been per-formed. Therefore, a pavement in state 3 cannot go up (im-prove) to state 2 after one year of service. Hence, P32 ¼ 0.Similar conclusions can be extended to other cells. Namely,the cells below the diagonal line shown in Table 14 will bezero and you can just leave them blank if you wish.

4. Please fill the whole table in a similar way. Remember that tsum of each row should be equal to 100.

5. When you complete a table for a pavement group, please selectthe next pavement group that you are most familiar with andfill out the table with the previously noted procedure.

Note that each cell in the tables presents the number ofpavements out of 100 that are now in state i that you expectin state j at the end of one year. Remember that one table isneeded for each pavement group. We suggest that you begin byselecting the pavement group that you believe you have had themost experience with.

Table 13. Different Pavement Groups

Group Pervious concrete thicknessa Vehicle trafficb Pavement agec Climate conditiond

1 Thin Light Primary interval Hard wet freeze

2 Thin Light Secondary interval Hard wet freeze

3 Thick Light Primary interval Hard wet freeze

4 Thick Light Secondary interval Hard wet freeze

5 Thick Heavy Primary interval Hard wet freeze

6 Thick Heavy Secondary interval Hard wet freezeaPervious concrete thickness: Thin ¼ 100mm < H ≤ 150mm (4 in: < H ≤ 6 in:); Thick ¼ 150mm < H < 250mm (6 in: < H < 10 in:).bVehicle traffic: Light = A pavement section that is generally exposed to ordinary cars, vans, and trucks. Heavy = A pavement section that is essentiallyexposed to heavy vehicles such as a plant that is frequently exposed to heavy vehicles with more than 25 average daily truck traffic. The heavy vehicle islimited to trucks with at least six wheels excluding panel and pickup trucks.cPavement Age: First Interval ¼ 1year < T ≤ 2year; Second Interval ¼ 2year < T < 5year.dCertain wet freeze areas that undergo a number of freeze-thaw cycles annually (15þ) and there is precipitation during the winter in which the ground staysfrozen as a result of a long, continuous period of average daily temperatures below freezing are referred to as hard wet freeze areas. These areas would havesituations in which the pervious concrete becomes fully saturated.

Table 14. Sample Transition Probability Matrix

PCDI

Future condition


8–10 6–8 4–6 2–4 0–2


Table 15. Group 1, Thickness: Thin/Vehicle Traffic: Light/Pavement Age: Primary Interval

PCDI

Future condition


8–10 6–8 4–6 2–4 0–2

Present condition State 1 (Very good) 8–10 p55 p54 — — —State 2 (Good) 6–8 — p44 p43 — —State 3 (Fair) 4–6 — — p33 p32 —State 4 (Poor) 2–4 — — — p22 p21


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Transition Probability Matrices for Six Groups of Pervious Concrete Parking Lots

Table 20. Group 6, Thickness: Thick/Vehicle Traffic: Heavy/Pavement Age: Secondary Interval

PCDI

Future condition


8–10 6–8 4–6 2–4 0–2


Table 16. Group 2, Thickness: Thin/Vehicle Traffic: Light/Pavement Age: Secondary Interval

PCDI

Future condition


8–10 6–8 4–6 2–4 0–2


Table 17. Group 3, Thickness: Thick/Vehicle Traffic: Light/Pavement Age: Primary Interval

PCDI

Future condition


8–10 6–8 4–6 2–4 0–2


Table 18. Group 4, Thickness: Thick/Vehicle Traffic: Light/Pavement Age: Secondary Interval

PCDI

Future condition


8–10 6–8 4–6 2–4 0–2


Table 19. Group 5, Thickness: Thick/Vehicle Traffic: Heavy/Pavement Age: Primary Interval

PCDI

Future condition


8–10 6–8 4–6 2–4 0–2



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Appendix II.

Table 21. Transition Probability Matrix for Group 1

PCDI

Future condition


8–10 6–8 4–6 2–4 0–2

Present condition State 1 (Very good) 8–10 0.61 0.39 — — —State 2 (Good) 6–8 — 0.62 0.38 — —State 3 (Fair) 4–6 — — 0.60 0.40 —State 4 (Poor) 2–4 — — — 0.58 0.42


PCDI

Future condition


8–10 6–8 4–6 2–4 0–2



PCDI

Future condition


8–10 6–8 4–6 2–4 0–2



PCDI

Future condition


8–10 6–8 4–6 2–4 0–2



PCDI

Future condition


8–10 6–8 4–6 2–4 0–2



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Table 30. Standard Deviation Matrix for Group 4

PCDI

Future condition


8–10 6–8 4–6 2–4 0–2



PCDI

Future condition


8–10 6–8 4–6 2–4 0–2



PCDI

Future condition


8–10 6–8 4–6 2–4 0–2



PCDI

Future condition


8–10 6–8 4–6 2–4 0–2

Present condition State 1 (Very good) 80–100 12 12 — — —State 2 (good) 60–80 — 10 10 — —State 3 (Fair) 40–60 — — 13 13 —State 4 (Poor) 20–40 — — — 18 18


PCDI

Future condition


8–10 6–8 4–6 2–4 0–2

Present condition State 1 (Very good) 8–10 12 12 — — —State 2 (good) 6–8 — 10 10 — —State 3 (Fair) 4–6 — — 12 12 —State 4 (Poor) 2–4 — — — 17 17


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PCDI

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http://dx.doi.org/10.1061/(ASCE)1076-0342(2002)8:4(139)

http://dx.doi.org/10.1061/(ASCE)1076-0342(2002)8:4(139)

http://dx.doi.org/10.1061/(ASCE)0733-947X(2006)132:2(141)

http://dx.doi.org/10.1061/(ASCE)0733-947X(1994)120:3(358)

Documents

Development of Pervious Concrete Pavement Performance Models Using Expert Opinions